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  "body": "\n## Definition\n\nLet $\\mathcal{E}$ be a [[topos|6837RS7N82]] with [[subobject classifier|NKP5TPQ79X]] $\\top \\colon 1 \\to \\Omega$, and write $PX$ for the [[exponential object|RHWD8DY41R]] $\\Omega^X$.\nThe *internal language* (Mitchell–Bénabou language) of $\\mathcal{E}$ is the typed language described as follows.\n\n1. The types are the objects of $\\mathcal{E}$; the type of a [[product|S5744QZV4N]] $X \\times Y$ is the product type, and $PX$ is the power type of $X$.\n2. There are variables of each type, and each morphism $f \\colon X \\to Y$ is a function symbol from type $X$ to type $Y$.\n3. Terms are built from variables by function symbols and by pairing $\\langle \\sigma, \\tau \\rangle$.\nFor terms $\\sigma, \\tau$ of the same type, $\\sigma = \\tau$ is a term of type $\\Omega$.\nFor $\\sigma$ of type $X$ and $\\tau$ of type $PX$, the membership $\\sigma \\in \\tau$ is a term of type $\\Omega$.\nFor a term $\\varphi$ of type $\\Omega$, the comprehension $\\set{x \\in X \\mid \\varphi}$ is a term of type $PX$.\n4. A *formula* is a term of type $\\Omega$.\n\nA term $\\tau$ of type $Y$ whose free variables are $x_1 : X_1, \\dots, x_n : X_n$ is interpreted as a morphism $X_1 \\times \\cdots \\times X_n \\to Y$.\nVariables are interpreted by projections, function symbols by composition, $\\sigma = \\tau$ by the characteristic morphism of the diagonal $Y \\to Y \\times Y$, and $\\sigma \\in \\tau$ by the evaluation morphism $X \\times PX \\to \\Omega$.\nA formula $\\varphi$ is *valid* in $\\mathcal{E}$ when its interpretation equals $\\top$ composed with the unique morphism $X_1 \\times \\cdots \\times X_n \\to 1$.\n\n## Remarks\n\nThe connectives $\\wedge, \\vee, \\Rightarrow, \\neg$ and the quantifiers $\\forall, \\exists$ are definable from equality, membership and comprehension; on subobjects they act as the Heyting algebra operations and as the adjoints to [[pullback|Q05K04QJ71]].\nThe valid formulas are closed under the rules of higher-order intuitionistic logic, but not in general under the law of excluded middle.\nExample: in $\\mathbf{Set}$, a formula is valid exactly when it is true for all values of its variables in the usual sense.\nExample: a morphism $f \\colon X \\to Y$ is a monomorphism exactly when the formula $\\forall x, x' \\in X\\, (f(x) = f(x') \\Rightarrow x = x')$ is valid.\nNon-example: in the topos $\\mathrm{Sh}(\\R)$ of sheaves on the real line, the formula $\\forall p \\in \\Omega\\, (p \\vee \\neg p)$ is not valid, so excluded middle is not part of the internal logic.\n",
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